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DC (NF/CGW) 48828

UCLES 2011 [Turn over

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSInternational General Certificate of Secondary Education

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ADDITIONAL MATHEMATICS 0606/21

Paper 2 October/November 2011

2 hoursCandidates answer on the Question Paper.

Additional Materials: Electronic calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all the questions.Give non-exact numerical answers correct to 3 significant figures, or 1 decimalplace in the case of angles in degrees, unless a different level of accuracy is

specified in the question.The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question orpart question.The total number of marks for this paper is 80.

For Examiners Use

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Total


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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax2+ bx + c = 0,

xb b ac

a=

2 4

2.

Binomial Theorem

(a + b)n = an + (n

1 )an1b + (

n

2 )an2b2 + + (

n

r)anrbr+ + bn,

where n is a positive integer and ( nr) = n!(n r)!r! .

2. TRIGONOMETRY

Identities

sin2A + cos2A = 1

sec2A = 1 + tan2A

cosec2A = 1 + cot2A

Formulae forABC

asinA

=b

sinB=

csin C

a2 = b2 + c2 2bc cosA

=1

2bc sinA


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1 Solve the equation |4x 5| = 21. [3]

2 Given that the straight line y = 3x + c is a tangent to the curve y =x2 + 9x + k, express kinterms ofc. [4]


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3

3 2

2+3

Without using a calculator, find the value of cos, giving your answer in the forma + b 3

c, where

a, b and c are integers. [5]


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4 (i) Given that y =1

x2 + 3, show that

dy

dx=

kx(x2 + 3)2

, where kis a constant to be found. [2]

(ii) Hence find 6x(x2 + 3)2 dx and evaluate 3

1

6x(x2 + 3)2

dx. [3]


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5 (a) The functions f and g are defined, forx by ,f :x 2x + 3,g :x x2 1.

Find fg(4). [2]

(b) The functions h and k are defined, forx > 0, byh :x x + 4,

k :x x . Express each of the following in terms of h and k.

(i) x x + 4 [1]

(ii) x x + 8 [1]

(iii) x x2 4 [2]


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6 Solutions to this question by accurate drawing will not be accepted.

The pointsA(1, 4),B(3, 8), C(13, 13) andD are the vertices of a trapezium in whichAB isparallel toDCand angleBAD is 90. Find the coordinates ofD. [6]


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7 (a) Given that tanx =p, find an expression, in terms ofp, for cosec2x. [3]

(b) Prove that (1 + sec)(1 cos) = sin tan. [4]


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8

A X

B

P

b

a

O

In the diagram OA = a, OB = b and AP =2

5AB .

(i) Given that OX = OP, where is a constant, express OX in terms of, a and b. [3]

(ii) Given also that AX = OB , where is a constant, use a vector method to find the value ofand of. [5]


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9 The table shows experimental values of two variablesx andy.

x 1 2 3 4 5

y 3.40 2.92 2.93 3.10 3.34

It is known thatx andy are related by the equation y = ax

+ bx, where a and b are constants.

(i) Complete the following table.

x x

y x

[1]

(ii) On the grid on page 11 ploty x against x x and draw a straight line graph. [2]

(iii) Use your graph to estimate the value ofa and ofb. [3]

(iv) Estimate the value ofy whenx is 1.5. [1]


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2

1

2

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O 4 6 8 10 12x

x

yx


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10 It is given that A = 3 21 5 and B = 1 42 3. (i) Find 2A B. [2]

(ii) Find BA. [2]


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(iii) Find the inverse matrix, A1. [2]

(iv) Use your answer to part (iii) to solve the simultaneous equations 3x + 2y= 23, x 5y = 19.

[2]


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11 (a) (i) Solve52x+3

252x=

252x

125x. [3]

(ii) Solve 1gy + 1g(y 15) = 2. [4]


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(b) Without using a calculator, and showing each stage of your working, find the value of

2 log12

4 1

2log

1281 + 4 log

123. [3]


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12 Answer only one of the following two alternatives.

EITHERy

xA

P(1,1n2)y =1n(x+1)1nx

CO

B

D

The diagram shows part of the curve y = 1n (x +1) 1nx. The tangent to the curve at the pointP(1, 1n 2) meets thex-axis atA and they-axis atB. The normal to the curve atPmeets thex-axis at Cand they-axis atD.

(i) Find, in terms of 1n 2, the coordinates ofA,B, CandD. [8]

(ii) Given thatArea of triangleBPDArea of triangleAPC

=1k

, express kin terms of 1n 2. [3]

OR

A curve has equation y =xex. The curve has a stationary point atP.

(i) Find, in terms of e, the coordinates ofPand determine the nature of this stationary point. [5]

The normal to the curve at the point Q (1, e) meets thex-axis atR and they-axis at S.

(ii) Find, in terms of e, the area of triangle ORS, where O is the origin. [6]


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Start your answer to Question 12 here.

Indicate which question you are answering. EITHER

OR

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Continue your answer here if necessary.

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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Everyreasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the

publisher will be pleased to make amends at the earliest possible opportunity.

University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of

Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

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